Throughput, Risk, and Economic Optimality of Runway Landing Operations

Transcription

1 Throughput, Risk, and Economic Optimality of Runway Landing Operations Babak Jeddi John Shortle Center for Air Transportation Systems Research George Mason University July 3, 27 7 th USA/Europe ATM 27 R&D Seminar

2 Runway Data Runway Occupancy Time (sec) Simultaneous Runway Occupancy Landing Time Interval (sec) What is the safe capacity of this runway? 2 Jeddi, B., J. Shortle, L. Sherry. 26. Statistics of the approach process at Detroit Metropolitan Wayne County Airport. International Conference on Research in Air Transportation. Belgrade, Serbia & Montenegro.

3 Data Source DTW (Detroit) multilateration data 1 Week: 2-Feb-23 thru 8-Feb-23 All runways (mostly 21L, 22R) Data restricted to Peak-time landings (> 7 arrivals per qtr-hr) IMC conditions Aircraft weight category pairs with 3 nm separation (e.g., Large-Large) Results of this talk can be generalized to varied fleet mix and weather conditions 3 Jeddi, B., J. Shortle, L. Sherry. 26. Statistics of the approach process at Detroit Metropolitan Wayne County Airport. International Conference on Research in Air Transportation. Belgrade, Serbia & Montenegro.

4 Capacity Capacity: Maximum achievable throughput on average Separation requirement: S time units Assume No gaps in arrival process Arrivals are separated by exactly S S S Capacity = 1 / S Example: S = 9 seconds, Capacity = 4 / hr Problem: Separation standard not always met 4

5 A Revised Definition S = Separation 16 Standard 14 (notional) Number Probability 6 Of Standard 4 Violation 2 T = Target Separation Time Separation (sec) Observed Time Separation Choose target separation T so that probability of separation violation is less than some small value. Restrict observations to peak periods Capacity = 1 / Expected Separation Buffer-adjusted capacity Jeddi, B., J. Shortle, L. Sherry. 26. Statistical separation standards for the aircraft approach process. Proceedings of the 25 th Digital Avionics Systems Conference, Portland OR, 2A1-2A1-13.

6 Runway Incursion-Based Capacity Determine target separation so that P{LTI < ROT} < α Shift LTI distribution to the left or right Example: for α = 1-4, increase separation by 15 sec.5.4 ROT ~ Beta( 25, 11; 6.1, 15.4 ) LTI ~ Gamma( 4; 11, 6 ) Target separation from 95 to 11 sec PDF Runway Occupancy Time (ROT) Landing Time Interval (LTI) Time (s) 6 Jeddi, B., J. Shortle, L. Sherry. 26. Statistical separation standards for the aircraft approach process. Proceedings of the 25 th Digital Avionics Systems Conference, Portland OR, 2A1-2A1-13.

7 Risk vs. Throughput Use different safety thresholds α to evaluate risk versus throughput.7.6 Runway Related Risk vs. Throughput RISK = P ( LTI < ROT ) Risk Prob(LTI < ROT) Throughput THROUGHPUT (Arrivals per quarter per Qtr-Hr) hour) 7 Jeddi, B., J. Shortle, L. Sherry. 26. Statistical separation standards for the aircraft approach process. Proceedings of the 25 th Digital Avionics Systems Conference, Portland OR, 2A1-2A1-13.

8 A Risk-Free Capacity Definition 2 Number Assume These 6 Are Go-Arounds 4 T = Target Separation Values are notional Assume the system is completely safe Simultaneous runway occupancy (SRO) is eliminated by go-around Assume pilot always takes go-around to avoid SRO (perfect information & execution) Time Separation (sec) Observed Time Separation

9 Trade-offs Number Probability 6Of 4 Go-Around 2 T = Target Separation Values are notional Time Separation (sec) Observed Time Separation Lowering the target separation allows more aircraft to land However, this also increases the rate of go-arounds At some spacing T, a maximum throughput is achieved Capacity = 1 / Expected Separation 9

10 Simultaneous Runway Occupancy P{ SRO} = = P P { LTI < ROT & Trailing aircraft lands} { Trailing aircraft lands LTI < ROT} P{ LTI < ROT} Enforced go-around = Zero { LTI < ROT} = P{Go Around} p P = P{SRO}= Zero LTI: ROT: SRO: 1 Landing Time Interval Runway Occupancy Time Simultaneous Runway Occupancy

11 Landing and Go-around Process Runway Go-Around p ω p ω λ=(1-p) ω ω (= p ω + λ) λ λ = rate of successful landings / h Go-Around ω = rate of attempts / h p ω = rate of go-around λ = rate of new arrivals 11 Goal: Maximize λ(ω) = [1-p(ω)] ω

12 Assumptions Distribution of time-separation unchanged along approach LTI and ROT of a lead-follower pair are independent Shifting LTI distribution to left or right does not change its shape Go-around is executed with perfect information 12

13 Maximizing Throughput 4.35 Successful Landings per Hour (λ) Lambda ( successful landing / h ) λ = Prob. of Go- Around (p) w (attempt / h) Landing Attempts per Hour (ω) 13

14 SRO and Wake Constraints Simultaneous Runway Occupancy.5.4 ROT ~ Beta( 25, 11; 6.1, 15.4 ) LTI ~ Gamma( 4; 11, 6 ).16 Wake Vortex Hazard LTI ~ Gamma( 4; 11, 6 ) P D F x Time (s) Prob{Go-around to avoid SRO} = Prob{LTI < ROT} Underlying model structure is same Different constraints yield different functions: Prob{Go-around} = p(ω) Time ( s ) Prob{Go-around to avoid wake hazard} = Prob{LTI < x }

15 With Wake Vortex Constraint Objective: Maximize λ(ω) = [1-p(ω)] ω Assume wake separation requirement x = 65 sec Successful Landings per Hour (λ) g( w, r ) λ(ω) Lambda( w ) maximal points Landing Attempts w (attempt per / h) Hour (ω)

16 Economic Optimality Definitions R: dollar benefit of a successful landing for all beneficiaries C: expected average cost of a go-around Maximize ES ( ω; R, C) = [ 1 p( ω) ] ω R p( ω) ω C 16 Illustration: For DTW distributions under IMC, 3 nmi sep. Without wake constraint C held constant ES (ω) ( 1$ / h ) C = $4, R = $1, 16 R = $2, 14 $ Benefit per hour C/R = ω ( attempts / h ) Landing Attempts per Hour (ω) R = $4, C/R = 1 C/R = 2

17 Optimal Capacity 35 Landings Equivalents per Hour g( ω; g( w, C/R) r ) (λ) (Assume R=1) Lambda( w ) maximal points C/R =1 C/R =2 C/R = 1 C/R = x = 65s C/R 1 ω * λ* P* % C/R =4 C/R = 3 C/R = w (attempt / h) ω (attempt / h) (with wake constraint) 17

18 Dependency on Cost / Revenue Optimal Throughput ( Landing/h ) Landing Capacity r = C/R x=65 x=7 x=75 Wake Constraint = 65 sec =7 sec =75 sec 18

19 Acknowledgments Wayne Bryant, Ed Johnson, NASA This talk solely represents the opinions of the authors 19

20 Summary Optimization model to maximize (without risk) Throughput Economic benefit Definition of capacity that Takes into account statistical variation of arrival process Does not depend on defining a safety level (e.g., P(SRO) < 1-5 ) Models are notional and demonstrate principles Models generalize to non-uniform fleet mix Potential applications Show capacity resulting from new technology (e.g., smaller variance in LTI) Relative benefits of addressing wake technology and constraints vs. runway occupancy constraint 2

21 21 Backup Slides

22 22 Time Separations

23 Detroit Airport Detroit Airport (DTW) 23

24 24 Multi-lateration Data Collection

25 Sample Collection Process Threshold Airplane i Airplane i+1 Runway Aircraft Type Threshold Leave Runway Heavy 1:23:14 1:24:4 Large 1:24:28 1:25:13 Large 1:26:16 1:27:12 Small 1:28:32 1:29:

26 Arrival Rates in every quarter hour, Runway 21L --*-- Arrival rate I M C periods >7` 26

27 Total Observations in peak periods Runway a/c Type 3L 3R 4L 4R 9L 9R 21L 21R 22L 22R 27L 27R Total % Not Available Small Large B Heavy Total landings, 2 Feb 3 8 Feb 3 on all twelve runways 1862 in periods with arrival rate per quarter hour >= 7 (peak periods) 27

28 Comparison with ASPM Rates 3 25 Arrival Rates ASPM & Observed CATSR ASPM # arrival/15 min Hours (Feb 1,3 to Feb 8,3) Average Difference:.24 arrivals / qtr-h 1 ASPM Observed (per quarter hour) Arrival rate comparison: CATSR minus ASPM Standard Deviation: 1.7 arrivals / qtr-h difference # Total difference: 16 landings or 3.6% -5-1 Hours (Feb 1,3 to Feb 8,3) 28 ASPM: Aviation System Performance Metrics

29 Lead-Follow Mixes Percentage (out of 185 pairs) Follow \ Lead Small Large B757 Heavy Sum Small Large B Heavy Sum

30 Separation Minima Standards ILS Approach In-Trail Threshold Separation Minima (nm) 1 Follow\ Lead Small Large B757 Heavy Small Large B Heavy class 6nm class 5nm class 4nm class 3nm 1) Ref: FAA Separation Rules For Arrivals and departures 3

31 Landing Time Interval (LTI) Gamma 31 LTI over the runway threshold Instrument meteorological condition (IMC) 3 nm pairs 523 samples (during IMC peak periods) Fit: Gamma(4;11,6): mean 16 sec, std. dev. 27 sec.

32 Inter-Arrival Distance (IAD) Gamma 32 IAD to the runway threshold Instrument meteorological condition (IMC) 3 nm pairs 523 samples (during IMC peak periods) Fit: Gamma(1.5;.35,6): mean 3.6 nm, std. dev..86 nm.

33 Independence of LTI One-Lag Scatter Plot.One-lag correlation coefficient:.25 Correlation coefficients for higher degrees of lags are smaller With some compromise, we decide the samples are independent In similar manner, we accept sample independence for IAD. Landing Time Interval (s) 33

34 Runway Occupancy Time (ROT) samples for all aircraft types, peak IMC periods Sample mean 49.1 s, standard deviation 8.1 s Beta(6.1,15.4) in the (25,11)s N(49, ) is rejected in the.1 significance level

35 ROT: IMC vs. VMC.3.25 VMC; Avg=5 Std=9 I MC; Avg=49, Std=8 Proportion ROT (s) 35 ROT for runways 21L/3R and 22R/4L IMC (59 samples), VMC (895 samples) No significant difference between IMC and VMC observed

36 Price of Risk Assume DTW peak period IMC distributions and the safe WV separation of 6 s. ω (attempt/h) λ(landing/h) p 7.4% 15% 25.6 % Risked (landing/h) In $ terms: Multiply the risky landings by R For example, 2.7R if 37.3 is pushed without go-around! 36

37 Wake Vortex Cost = 6.1 Landing/hour 6.1 x 1 h/day x 365 day ~= 22, Landing/h (for a moderately busy runway!) 22, x 2 runway x 35 airport ~= $1, x 1,56, ~= $1.6 b 1,56, landing/year 37

38 38 Generalized Model ( ) ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) = = = ω ω ω λ ω ω ω λ ω ω ω ω ω p R C R C p R C p R p C R ES 1, ; Max. Let: ( ) ( ) ω ω ω λ ω p R C R C g = ; ( ) R C Maximize g C R ES Maximize ;, ω; ω ω ω Then:

39 Model II: properties of Optimal Solution Properties: 1. For a given C and R, g(ω; C/R) are uni-modal 2. g(ω;c/r) decreases as C/R increases for any fixed ω 3. C C C g ω; = λ( ω) ω p( ω) λ( ω) as R R R 4. ω*(c/r) = Argmax{g(ω;C/R)} is decreasing in C/R for 28<ω<Argmax{dp/dω} g(ω;c/r) g ( ω ) r = C/R 4 = 4 r = C/R 2 = 2 r = C/R 1 = 1 λ C/R (r = = ) ω ( attempt / h ) g(ω;c/r) g( w, r ) Lambda( w ) maximal points C/R =2 C/R =4 C/R = w (attempt / h) With WV effect 39 Without WV effect

40 Comparison of optimal ES and g 4 38 C/R r = 4= 4 C/R r = 2= 2 C/R r = 1= 1 C/R λ (r = ) 36 4 g ( ω ) g(ω;c/r) ES (ω) ( 1$ / h ) C = $4, R = $1, R = $2, R = $4, 12 ω ( attempt / h ) ω ( attempts / h )

41 Only SRO aircraft / h r ω*(r) λ*(r) 41

42 Throughput in WV threshold 4 38 Lambda = X Omega = X ( aircraft / h ) Wake Vortex Threshold, X, ( s ) 42

43 Optimal solution in terms of WV threshold p*.78 wv thr (nmi) 1.67 wv thr (s) 5 w* 4. L* p*.77 p^ w (attempt / h) p* w v threshold (s) 43

44 Cost of Wake Vortex (in # of landings) WV exists! The difference between the optimal λ with WV threshold and without any WV threshold provides the answer. Example: If the absolute safe WV threshold is 6s then WV cost = = 5.1 landing/h 44